Adviser: Frank, Yeong-Sung Lin Present by Sean Chou

1. Optimal Replacement and Protection Strategy for Parallel SystemsRuiPeng, Gregory Levitin, Min Xie and SzuHui Ng Adviser: Frank, Yeong-Sung Lin Present by Sean Chou

2. Agenda • Introduction • Problem Formulation and Description of System Model • Optimization Technique • Illustrative Example • Conclusions

3. Agenda • Introduction • Problem Formulation and Description of System Model • Optimization Technique • Illustrative Example • Conclusions

4. Introduction • When a system is subjected to both internal failures and external impacts, maintenance and protection are two measures intended to enhance system availability. • For multistate systems, the system availability is a measure of the system’s ability to meet the demand (required performance level). • In order to provide the required availability with minimum cost, the optimal maintenance and protection strategy is studied.

5. Introduction • For systems containing elements with increasing failure rates, preventive replacement of the elements is an efficient measure to increase the system reliability (Levitinand Lisnianski 1999). • Minimal repair, the less expensive option, enables the system element to resume its work after failure, but does not affect its hazard rate (Beichelt and Fischer 1980; Beichelt and Franken 1983).

6. Introduction • In order to increase the survivability of a componentunder external impacts, defensive investments can be made to protect the component. • It is reasonable to assume thatthe external impact frequency is constant over time and that the probability of thecomponent destruction by the external impact decreases with the increase of theprotection effort allocated on the component.

7. Introduction • We consider a parallel system consisting of components with different characteristics(nominal performances, hazard functions, protection costs, etc.). • Theobjective is to minimize the total cost of the damage associated with unsupplieddemand and the costs of the system maintenance and protection. • A universalgenerating function technique is used to evaluate the system availability for anymaintenance and protection policy. • A genetic algorithm is used for the optimization.

8. Agenda • Introduction • Problem Formulation and Description of System Model • Optimization Technique • Illustrative Example • Conclusions

9. Problem Formulation and Description of System Model • Assumptions: • All N system components are independent. • The failures caused by the internal failures and external impacts are independent. • The time spent on replacement is negligible. • The time spent on a minimal repair is much less than the time between failures.

10. Problem Formulation and Description of System Model • A system that consists of N components connected in parallel is considered. • The lifetime for the system is denoted as Tc. • For each component i, its nominal performance is denoted as Gi • The expected number of internal failures during time interval (0, t) is denoted as λi(t)

11. Problem Formulation and Description of System Model • Each component is subjected to internal failures and external impacts. • Two maintenance actions (Sheuand Chang 2009) • Preventive replacement • The ith component is replaced when it reaches an age Ti. • Minimal repair • This action is used after internal failures or destructive external impacts and does not affect the hazard function of the component. • The average cost • σi internal failure • θi external impact • The average time • ti internal failure • τi external impact

12. Problem Formulation and Description of System Model • The average number of internal failures during the period between replacements λi(Ti) can be obtained by using the replacement interval Ti for each element. • The number of preventive replacements ni during the system life cycle: • The total cost of the preventive replacements of component i during the system life cycle is:

13. Problem Formulation and Description of System Model • Therefore, the total expected number of internal failures of the component iduring the system life cycle is:

14. Problem Formulation and Description of System Model • We use xi to denote the protection effort allocated on component i and ai to denotethe unit protection effort cost for component i. • The total cost of component iprotectionis aixi. • It is assumed that the external impact frequency q is a constant. In thiscase the expected number of the external impacts during the system lifetime is qTc. • The expected impact intensity is d.

15. Problem Formulation and Description of System Model • The component vulnerabilityis evaluated usingthe contest function model (Hausken 2005; Tullock 1980; Skaperdas1996): • The expected number of the failurescaused by the external impacts is therefore

16. Problem Formulation and Description of System Model • Hausken and Levitin (2008) discussed the meaning of the contest intensityparameter m. • m = 0, the success probability of the two parties is random generated. • 0 < m < 1, there is low Marginal Effect that the amount of resources have little effect on the probability of success. • m = 1, the investment has proportional impact on the vulnerability. • m > 1, the relative between resource and vulnerability is show as the exponential grow. • m = ∞, It would definitely win the competition as long as one side invests a little more than the other.

17. Problem Formulation and Description of System Model • The total expected repair time of component iis • The expected minimal repair cost of component iis • The availability of each element is

18. Problem Formulation and Description of System Model • The system capacity distribution must be obtained to estimate the entire system availability. • It may be expressed by vectors G and P. • G = {Gv} is the vector of all the possible total system capacities, which corresponds to its V different possible states • P = {pv} is the vector of probabilities, which corresponds to these states.

19. Problem Formulation and Description of System Model • If we denote the system demand as W, the unsupplied demand probability: • The reliability of the entire system requires an availability index A = 1 – Pudthat is not less than some preliminary specified level A*. • The total unsupplied demand cost:

20. Problem Formulation and Description of System Model • The general formulation of the system replacement versus protection optimization problem can be presented as follows: • Find the replacement intervals and protection efforts for system element T = (T1, T2, …, TN) and x = {x1, x2, …, xN} that minimize the sum of costs of the replacement, protection and unsupplied demand:

21. Agenda • Introduction • Problem Formulation and Description of System Model • Optimization Technique • Illustrative Example • Conclusions

22. Optimization Technique • In this work, the genetic algorithm (GA) is used to solve the optimal (T, x). • First, an initial population of Ns randomly constructed solutions (strings) is generated. • Within this population, new solutions are obtained during the genetic cycle by using crossover, and mutation operators. • The crossover produces a new solution (offspring) from a randomly selected pair of parent solutions, facilitating the inheritance of some basic properties from the parents by the offspring.

23. Optimization Technique • The simple operator should be used to evaluate the probability that the randomvariable G represented by polynomial u(z) defined in (9.12) does not exceed thevalue W: • Furthermore the availability index A of the entire system can be obtained as • The total unsupplied demand cost can be estimated as

24. Optimization Technique • Solution Representation and Decoding Procedures • Each solution is represented by string S = {s1, s2, …, sN}, where sicorresponds to component i for each i = 1, 2, …, N. • Each number si determines both the replacement interval of component i (Ti) and the protection effort allocated on component i (xi). • To provide this property all the numbers si are generated in the range

25. Optimization Technique • The solutions are decoded in the following manner: • For given vi and xi the corresponding si is composed as follows:

26. Optimization Technique • The possible replacement frequency alternatives are ordered in vector h = { h1, h2, …, hΛ } so that hi < hi+1, where hi represents the number of replacements during the operation period that corresponds to alternative i. • After obtaining vi from decoding the solution string, the number of replacement for component i can be obtained as • Furthermore replacement interval for component i can be obtained as

27. Optimization Technique • In order to let the genetic algorithm search for the solution with minimal total cost, when A is not less than the required value A*, the solution quality (fitness) is evaluated as follows: • For solutions that meet the requirements A ≧A*, the fitness of the solution is equal to its total cost.

28. Agenda • Introduction • Problem Formulation and Description of System Model • Optimization Technique • Illustrative Example • Conclusions

29. Illustrative Example • The system considered in this example consists of • Eight parallel components. • The lifetime of the system Tc is 120 months. • The system demand W is 100. • Replacement frequency alternatives Λ = 6 • The alternatives are h = {4, 9, 14, 19, 24, 29}. • Alternatives for replacement interval are 24, 12, 8, 6, 4.8 and 4 months. • For our example we assume that q = 0.5, d = 30, a = 3,000 and the maximum protection effort allowed to be allocated on a component is M = 50. • We have:

30. Illustrative Example • For a given solution string, S = {195 280 49 218 176 132 270 120} is decodedinto T = (6 4.8 12 8 8 24 24 24) and x = (32 46 8 36 29 22 45 20). • If we assumem = 1 and A* = 0.90, the fitness function for this solution takes the valueF(T, x) = 19,210.

31. Illustrative Example

32. Illustrative Example • The maximum availability A = 0.9592 can be achieved when maximal possibleprotection and replacement frequency are applied. • In this case all the componentsare replaced every 4 months and the protection effort on each component is 50. • The corresponding total cost is 35,334.

33. Agenda • Introduction • Problem Formulation and Description of System Model • Optimization Technique • Illustrative Example • Conclusions

34. Conclusions • This chapter considers a parallel system that can fail due to both internal failures and external impacts. • The availability of an element can be increased in two ways: • (1) replace the element more frequently • (2) allocate more resource to protect it against external impacts. • The maximum availability can be achieved when maximal possible protection and replacement frequency are applied.

35. Conclusions • Component failure • Defender choose network component base on hvi • Tc = Tfail • It would not happened component failure until the time interval Ti. After Ti, component failures happened interval would be “Poisson Arrival Process”.

36. Thanks for your listening.